Elementary proof of the estimation of the sum of fractional parts

Author(s):  A.V. Shutov, candidate of Sciences, associate Professor, Vladimir State University named after Alexander and Nikolay Stoletovs, Vladimir, Russia, a1981@mail.ru

Issue:  Volume 50, № 4

Rubric:  Physics. Mathematical modeling

Annotation:  Let  be an irrational number. Denote by { } n q a sequence of partial quotients of the continued fraction expansion of  and by { } n n P Q a sequence of partial convergents to  . Assume that 1 0 1 ( , ) ({ } ) 2 n n i С   i     . Estimates for ( , ) n С   are important both by themselves and in connection with their applications in a some number-theoretic problems, such that studying the remainder term of the distribution of the fractional parts of a linear function and in number-theoretic methods of approximate integration. The best known estimate of ( , ) n С   is 1 | ( , ) | k n i i С   C q    for 1 k  i Q . In this paper, we give a new short proof of this result.

Keywords:  sums of fractional parts, uniform distribution, continued fractions.

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